Module 3 Lecture - Transformations and Non-parametric Comparisons for Two Groups

Analysis of Variance

Quinton Quagliano, M.S., C.S.P

Department of Educational Psychology

1 Overview and Introduction

Agenda

1 Overview and Introduction

2 Solutions for Assumption Violations

3 Conclusion

1.1 Objectives

1.2 Introduction

2 Solutions for Assumption Violations

Agenda

1 Overview and Introduction

2 Solutions for Assumption Violations

3 Conclusion

2.1 3 General Strategies

  • Important: There is considerable controversy and conflicting ideas in this area, as to 'what should we do' - take non-parametric class for more fun!
  • We need a way to address assumption violations when they occur (well mostly - see Do Nothing!)
    • As mentioned before, the main risk of assumption violations, is that they reduce power and raise Type II error rate
    • Since we are coming into most studies with a hypothesis of differences existing, we want to plan an analysis that has the sufficient power to detect that hypothesized difference
  • We have 3 strategies for approaching assumption violations:

2.2 Do Nothing!

  • Without going into too much detail, we are concerned with how robust a test is, or how resilient a test is to assumption violations, and how well it works under less-than-ideal circumstances

  • Some researcher’s hold that many commonly used tests, i.e., t-tests are reasonably robust at baseline to assumption violations

    • This becomes even more true at large \(n\) size (See discussion in [Sample Size] section])
    • However, when samples are small or the groups being compared are unequal - this route cannot be recommended

2.3 Correct for Violations in Data (Transformation)

  • There are several options for selectively transforming and or trimming our data to correct for certain patterns of skewness, kurtosis, or outliers
    • However, be aware these are not panaceas - they come with their own issues

Transformations

  • Making mathematical variable transformations is largely used to address the [Normality Assumption], but maybe indirectly solve other issues as well

  • The exact transformation is dependent on the type of problem, specifically the skew:

    • Positive/right skew: Logarithmic (Severe) or Square Root (Moderate)
    • Negative/left skew: Reflect and Logarithmic (Severe) or Reflect and Square Root (Moderate)
  • Advantages:

    • Makes use of all available data
    • Allows for use of a traditional well-known technique.
  • Disadvantages:

    • Interpretability can be questionable
    • Fixing one assumption violation can create others

Trimming

  • Another option, particularly useful for negatively kurtotic (platykurtic) distributions (relatively flat distributions with an unusual number of observations in the tails) is to use variable trimming.

  • A trimmed sample is a sample where a fixed percentage of extreme values is removed from each tail.

    • Of course, if you are comparing groups, you would want to trim the same percentage from the tails of both groups to be fair.
    • 0.20 from each tail is the most common amount to trim
  • Another related option is using winsorizing

    • A Winsorized sample replaces the trimmed values by the most extreme value remaining in each tail.
    • Note: degrees of freedom for a test on a trimmed or Winsorized sample must be adjusted for the data trimming. For both, you would subtract your total N by the number of trimmed cases to get a new value for N. (We do this even for Winsorized samples b/c the added values are really pseudovalues.)
  • Advantages:

    • Allows for use of a traditional well-known technique.
    • Interpretability of variable remains intact
  • Disadvantages:

    • Loss of information - what if those outliers were an important part of the phenomenon

2.4 Use Non-parametric Tests

3 Conclusion

Agenda

1 Overview and Introduction

2 Solutions for Assumption Violations

3 Conclusion

3.1 Recap

3.2 Lecture Check-in

  • Make sure to complete any lecture check-in tasks associated with this lecture!

Module 3 Lecture - Transformations and Non-parametric Comparisons for Two Groups || Analysis of Variance